Ask a Mathematician!

[ParadigmShifter]

You need calculus to derive the Taylor Series for exp(z), sin(z), cos(z) though

EDIT: Since they are all solutions to differential equations...

 

[madviking]

You need calculus to derive the Taylor Series for exp(z), sin(z), cos(z) though

Then I don't know how to derive it without calc.

 

[ParadigmShifter]

Indeed. You also forgot to mention that you can only say exp(z) = cos(z) + i sin(z) since all 3 power series are absolutely convergent over C.

 

[ShahJahanII]

Makes sense, thank you!

 

[IdiotsOpposite]

Could you explain Euler's identity without calculus?

That depends. Do you know de Moivre's identity?

 

[madviking]

That depends. Do you know de Moivre's identity?

Isn't that derived from Euler's identity?

 

[IdiotsOpposite]

Isn't that derived from Euler's identity?

It can be, but it was proven earlier than Euler's identity, so de Moivre's Theorem can obviously be derived without it.

EDIT: Actually, I can prove it right now, by induction, if you wish.

 

[ParadigmShifter]

But it still involves calculus. Once you can write down the Taylor Series for sin, cos, exp, and notice how they are related, it is easy. You can't do that without calculus though.

EDIT: You CAN show that complex multiplication is a spiral enlargement though, without calculus. The tricky thing is getting the number pi (EDIT: and e) in there... which involves the trig Taylor Series.

 

[caketastydelish]

What books would you recommend for someone that is going to take trigonometry in the summer and is very rusty with math?

 

[madviking]

What books would you recommend for someone that is going to take trigonometry in the summer and is very rusty with math?

Any textbook will go into sufficient trigonometric detail. As for non textbooks, idk.

 

[IdiotsOpposite]

What books would you recommend for someone that is going to take trigonometry in the summer and is very rusty with math?

I don't. I recommend talking with someone and any trig textbook and working through it (with them to help you if you get confused). And don't forget to do at least some of the exercises. That's why they're there!

If you need a particular textbook, just PM me and I'll help you find one.